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11    THE SPACES L1 AND L2       169

(c)    Extend the results of Section 13.11 to the case where p is any number such that
1 < p < + oo.

(d)    Suppose that p>\. Let /e &&,g e && be two functions 2> 0, Show that we
have Ni(fg) = Np(/)Nq(#) if and only if there exist  two constants a > 0, /3 > 0
such that (x.(f(x))p = f$(g(x))q almost everywhere.

(e)    Suppose that the measure (JL is bounded and that ft(X) = 1 . For each r > 0
and each measurable function/^ 0 such that/1" is integrable, show that the mapping
/>h-»Np(/) is an increasing function on the interval ]0, r] (use Holder's inequality).

As/>->0, its limit is expf f   log \f\ dp\t or 0 if f   log \f\ d^= — oo. If the limit is

& 0, it follows that /(*) ^ 0 almost everywhere. If exp( f * log |/| dp\ = f |/| dp,,
the function |/| is constant almost everywhere.

(f)    Let /^ 0 be a /^-measurable function, and for each a > 0 let Aa be the set of all
x e X such that/(*) ;> a. If/ e JSfg, then apjLt(Aa) g J/p^. Conversely, if the measure

/A is bounded and if there exist constants C > 0, e > 0 such that /x(A«) ^ C • a~p~e
for all a> 0, then/e &£ (cf. Section 13.9, Problem 3).

13.   Let /be a real valued function ^0 defined on R* = ]0, +oo[, which is Lebesgue-
measurable and such that |/|p is Lebesgue-integrable (1 < p < -f oo).

(a)    The function /is integrable on every compact subset of [0, -f oo [, and in particular

rx

F6c) =     /(O dt is defined for all x > 0 (use Holder's inequality). As x tends to 0
Jo

or to + oo, the quotient F(x)/x(p~1)/p tends to 0 (same method).

(b)    Show that the function F(x)/x is /?th-power-integrable on ]0, -f oo [ and that

r+00/F(x)

Jo   -r

("Hardy's inequality"). (Consider first the case where /e Jf(Rf). For each compact

/.*
interval [a, b] <=• RJ , majorize the integral     (F(/)//)p dt, by integrating by parts and

Ja

using Holder's inequality.)

14.    Let p be a real number >0 and let / be a complex ft-measurable function such
that (1) |/|p is /x-integrable and (2) etf is ^-integrable for 0</</0. Show that
Np(/~1(^r/™ 1)— /)->0 as t-*Q. (Let /== u + w where u and v are real; observe
that if we put wt = etu — I — tu, we have wsjs g w,/t for 0 <l ^ ^ t. If 0 < p < 1, use
the elementary inequality (# -f 6)p ^ ap -f ^p when <7, b are ^0. Finally, apply (13.8.1)
and (13.8.4).)

15.    Let (Xn)nzi be a strictly increasing sequence of integers >0, and let h be a rational
integer. For each integer N ^ 1 , put

Show that the series.   For every real number p such that 0 < p < +00, and every mapping /: X ->R, put